Ecological Archives
E086-055-A7

Appendix G. More on caveats
on the use of the simple model. A pdf file of this
appendix is also available.

In the strict two component lottery,
the fluctuation measure is inversely proportional to the square of the population
density, for two species with equal lifetimes. Since

and from (A.3)

if

(G.1)

where is
relative abundance. This prediction and that of strict antiphase are affected
by the considerations below.

Age effects – There are in
these data no trees whose ages are known with precision: the conversion of size
profiles to age profiles is inevitably imprecise. The lack of precise age data
has implications for working out strict antiphase. It may have some importance
for the relation between and
. An incorrect scaling factor
would have no effect on . However,
the relation between age and size must have dispersion and this dispersion increases
with age. Such dispersion will reduce both and
but is not likely to affect their ratio.

is
thus unlikely to be affected greatly by lack of accurate age information. The
relation between and may
however be affected, for if the mean lifetimes of two species are different,
(DA7.1) becomes

(G.2)

Finally, the measure of fluctuations
has for each species been calculated about an exponential with mean lifetime
equal to that of the sample. Deviations from any other exponential would be
greater.

Statistical effects – In
a strict two component lottery statistical fluctuations in recruitment would
be binomial and because either one species or the other is recruited to each
newly vacant site statistical fluctuations which occur when both species have
recruitment ON would preserve strict antiphase. This is no longer necessarily
true when size profiles are drawn from samples with different numbers of (potential)
homes. Fluctuations would then not necessarily exhibit strict bin by bin antiphase,
but one would expect the overall pattern to be preserved.

Even in a strict two component lottery,
statistical fluctuations can affect the value of the measure .
For a rare species, such fluctuations have a Poisson variance; for two species
of relatively similar abundance the binomial fluctuations are correlated. For
the species in Fig. 2 of the main text, the square root of the variance V()
estimated in Appendix E, Eq. E.7 is ~ 0.5 (larger in the case of J.
chamelensis), where the values of span
the range 0.5 – 5.0.

Temporal variation of the number
of homes – If the number of homes available varies with time then (i) an
in-phase variation in recruitment is superimposed on the antiphase behaviour
and (ii) both species suffer a component of fluctuation in recruitment (by number)
which is the same fraction of mean recruitment for each.

Equation (A.4) would be replaced
by

(G.3)

where A(t) is the modulation
imposed on the two component lottery through the variation in the number of
homes available as a function of time.

Write recruitments
where (for equal lifetimes) the sum is
a constant, so that

(G.4a)

(G.4b)

The recruitment deviations are a
mixture of an in phase component ()
and a strictly out of phase component (Fig. G1).

If and
are not correlated, then

(G.5a)

(G.5b)

The ratio lies
between 1 (when dominates) and
(DA7.1) (when dominates).
Since a violently fluctuating species will scarcely have ,
when , will be small and easily
dominated by . It is no surprise
that relative abundances of 10:1 do not yield ratios
of 100, but the species with lower mean recruitment still has larger fractional
deviation. Finally, there remains the possibility of a correlation between
and .
As we have defined it, is the
increase in recruitment of over
. In our specific model (section
3), environmental effects reduce quasiperiodically
to ~ 0. It might well be that the same environmental effects depress the number
of homes available, which would introduce a positive correlation between and
. Then
would be increased over the ratio of (G.4a) to (G.5b).

FIG.
G1. Illustration of Eqs. G.4 and G.5. (a) A simple pattern of recruitment
fluctuations for a fixed number of sites. The upper histogram shows ,
the lower for
, and
. These
periodic fluctuations exhibit strict antiphase and =
9. (b) The same pattern of recruitment fluctuations as (a), but in this
case the number of sites varies with ,
and four
times the period of the recruitment to such sites as exist. Strict antiphase
has been lost and in this case, =
5.